A018910 -id:A018910 - cp's OEIS frontend

A121605 Primes of the form (Fibonacci[n+3] + 2) = A018910[n], Pisot sequence L(4,5).

5, 7, 23, 379, 17713, 102334157, 3416454622906709, 1100087778366101933, 468340976726457153752543329995931, 40934782466626840596168752972961528246149

Offset: 1

Alexander Adamchuk, Aug 09 2006

nonn

a(n) = Fibonacci[A121606[n] + 3 ] + 2. Indices n = {1,2,5,11,19,37,73,85,155,193,227,233,257,785,797,1277,2371,2771,...} corresponding to prime A018910[n] = (Fibonacci[n+3] + 2) are listed in A121606[n].

Cf. A121606, A018910, A048584.

Select[Table[Fibonacci[n+3]+2,{n,1,300}],PrimeQ]

A121606 Indices n such that A018910[n] = (Fibonacci[n+3] + 2) are primes.

Original entry on oeis.org

1, 2, 5, 11, 19, 37, 73, 85, 155, 193, 227, 233, 257, 785, 797, 1277, 2371, 2771, 4213, 9107

Offset: 1

Alexander Adamchuk, Aug 09 2006

nonn

A018910[n] is Pisot sequence L(4,5). Prime Pisot L(4,5) numbers are A018910[a(n)] = Fibonacci[a(n) + 3 ] + 2 = {5,7,23,379,17713,102334157,...} = A121605[n]. Most listed a(n) are prime except a(8),a(9),a(14),a(18).

Cf. A121605, A018910, A048584.

Do[f=Fibonacci[n+3]+2;If[PrimeQ[f],Print[n]],{n,1,3000}]
Select[Range[10000],PrimeQ[Fibonacci[#+3]+2]&] (* Harvey P. Dale, Jul 01 2019 *)

More terms from Harvey P. Dale, Jul 01 2019

A020717 Pisot sequences L(6,9), E(6,9).

Original entry on oeis.org

6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142

Offset: 0

David W. Wilson

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

Colin Barker, Table of n, a(n) for n = 0..1000
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for Pisot sequences
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Subsequence of A001611, A048577.
See A008776 for definitions of Pisot sequences.
Pairwise sums of A018910.

Table[Fibonacci[n + 5] + 1, {n, 0, 36}] (* Michael De Vlieger, Jul 27 2016 *)

pisotE(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
  a
}
pisotE(50, 6, 9) \\ Colin Barker, Jul 27 2016

a(n) = Fibonacci(n+5)+1 = A001611(n+5).
a(n) = 2*a(n-1) - a(n-3).
a(n) = A020706(n+1). - R. J. Mathar, Oct 25 2008

A020743 Pisot sequence L(7,10).

Original entry on oeis.org

7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143

Offset: 0

David W. Wilson

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for Pisot sequences
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Subsequence of A018910. See A008776 for definitions of Pisot sequences.

Cf. A020717.

LinearRecurrence[{2,0,-1},{7,10,15},40] (* Harvey P. Dale, Jun 10 2022 *)

pisotL(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
  a
}
pisotL(50, 7, 10) \\ Colin Barker, Aug 07 2016

a(n) = Fib(n+5)+2 = A157725(n+5). a(n) = 2a(n-1) - a(n-3).

A048584 Pisot sequence L(5,7).

Original entry on oeis.org

5, 7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143

Offset: 0

David W. Wilson

nonn

a(n)= BA^(n)B(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g. 5=`00`, 7=`010`, 10=`0110`, 15=`01110`,..., in Wythoff code.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Subsequence of A018910. See A008776 for definitions of Pisot sequences.

LinearRecurrence[{2,0,-1},{5,7,10},40] (* Harvey P. Dale, Oct 02 2016 *)

pisotL(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
  a
}
pisotL(50, 5, 7) \\ Colin Barker, Aug 07 2016

a(n) = Fib(n+4)+2. a(n) = 2a(n-1) - a(n-3).

a(n)=A020743(n-1), n>0. - R. J. Mathar, Oct 15 2008

A166876 a(n) = a(n-1) + Fibonacci(n), a(1)=1983.

Original entry on oeis.org

1983, 1984, 1986, 1989, 1994, 2002, 2015, 2036, 2070, 2125, 2214, 2358, 2591, 2968, 3578, 4565, 6162, 8746, 12927, 19692, 30638, 48349, 77006, 123374, 198399, 319792, 516210, 834021, 1348250, 2180290, 3526559, 5704868, 9229446, 14932333, 24159798

Offset: 1

Geoff Ahiakwo, Oct 22 2009

Starting at some a(1)=s and creating further terms with the recurrence a(n)=a(n-1)+A000045(n) defines a family of sequences with recurrences a(n)= 2*a(n-1) -a(n-3).
The generating functions are x*( s+(1-s)*x+(1-s)*x^2 )/((1-x) * (1-x-x^2)).
The terms are a(n) = A000045(n+2)+s-2 = s + A001911(n-1) = (2*s+1+k)/2 where k=A166863(n-1), n>=1.
Examples: Up to offsets, s=1 yields A000071, s=2 yields A000045 shifted left thrice, s=3 yields A001611 shifted left thrice, s=4 yields A018910.
I appreciate the editing by R. J. Mathar. However I would like further analysis of the following formula. The sequence which I call GAP can have any integer as its first term, not just 1983. Thus a(1) can be 0, 1, 2, 3,... Then a(2) is always a(1)+ 1, while a(3) is a(1) + k(n)/2; where k(n) = k(n-2)+ k(n-1)+4 (This is a separate sequence submitted for consideration). [Geoff Ahiakwo, Nov 19 2009]

			For s=1983, n=3, we have k= A166863(2)= 5, a(3) = (2s+1+k)/2 = (2*1983+1+5)/2 = 1986.
For n=3, a(3)= a(1)+ k(3)/2 = a(1)+ [K(3-2)+ k(3-1)]/2 + 2 = a(1)+ 1 + 2 thus if a(1)is 0, a(3)= 3; if a(1)= 5, a(3)= 8; if a(1)=1983, a(3)= 1986, etc. [_Geoff Ahiakwo_, Nov 19 2009]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1)

Cf. A001911, A000071

LinearRecurrence[{2, 0, -1}, {1983, 1984, 1986}, 100] (* G. C. Greubel, May 27 2016 *)

a(n) = 2*a(n-1) - a(n-3).
G.f.: x*(-1983 + 1982*x + 1982*x^2)/((1-x)*(x^2+x-1)).
Let a(n)= a(1)+ k(n)/2, then G.f.: k(n)= k(n-2)+ k(n-1) + 4. - Geoff Ahiakwo, Nov 19 2009

Definition and comments edited by R. J. Mathar, Oct 26 2009

A277084 Pisot sequence L(4,14).

Original entry on oeis.org

4, 14, 49, 172, 604, 2122, 7456, 26198, 92052, 323444, 1136489, 3993295, 14031289, 49301911, 173232725, 608689936, 2138761243, 7514991434, 26405516950, 92781386582, 326007088306, 1145495077635, 4024940008834, 14142480741305, 49692606865991, 174605518105877

Offset: 0

Ilya Gutkovskiy, Sep 29 2016

There is no simple g.f. - Ilya Gutkovskiy, May 23 2019

Ilya Gutkovskiy, Pisot sequences L(x,y)
Index entries for Pisot sequences

Cf. A008776 for definitions of Pisot sequences.
Cf. A010904 (Pisot sequence E(4,14)), A251221 (seems to be Pisot sequence P(4,14)).
Cf. A018910, A020706, A004119, A020707, A048582, A020734.

RecurrenceTable[{a[0] == 4, a[1] == 14, a[n] == Ceiling[a[n - 1]^2/a[n - 2]]}, a, {n, 25}]

a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 4, a(1) = 14.

A258316 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with each 2X2 subblock having clockwise pattern 0000 or 0011.

Original entry on oeis.org

5, 7, 7, 10, 9, 10, 15, 12, 12, 15, 23, 17, 15, 17, 23, 36, 25, 20, 20, 25, 36, 57, 38, 28, 25, 28, 38, 57, 91, 59, 41, 33, 33, 41, 59, 91, 146, 93, 62, 46, 41, 46, 62, 93, 146, 235, 148, 96, 67, 54, 54, 67, 96, 148, 235, 379, 237, 151, 101, 75, 67, 75, 101, 151, 237, 379, 612

Offset: 1

R. H. Hardin, Jun 29 2015

Table starts
...5...7..10..15..23..36..57..91.146.235.379.612..989.1599.2586.4183.6767.10948
...7...9..12..17..25..38..59..93.148.237.381.614..991.1601.2588.4185.6769.10950
..10..12..15..20..28..41..62..96.151.240.384.617..994.1604.2591.4188.6772.10953
..15..17..20..25..33..46..67.101.156.245.389.622..999.1609.2596.4193.6777.10958
..23..25..28..33..41..54..75.109.164.253.397.630.1007.1617.2604.4201.6785.10966
..36..38..41..46..54..67..88.122.177.266.410.643.1020.1630.2617.4214.6798.10979
..57..59..62..67..75..88.109.143.198.287.431.664.1041.1651.2638.4235.6819.11000
..91..93..96.101.109.122.143.177.232.321.465.698.1075.1685.2672.4269.6853.11034
.146.148.151.156.164.177.198.232.287.376.520.753.1130.1740.2727.4324.6908.11089
.235.237.240.245.253.266.287.321.376.465.609.842.1219.1829.2816.4413.6997.11178
Apparently: put 1s in some number of nonadjacent columns or put 1s in some number of nonadjacent rows

			Some solutions for n=4 k=4
..0..0..0..0..0....1..1..1..1..1....0..0..0..0..0....1..1..1..1..1
..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..0..0..0..0..0....1..1..1..1..1....0..0..0..0..0....0..0..0..0..0
..0..0..0..0..0....0..0..0..0..0....1..1..1..1..1....0..0..0..0..0
..1..1..1..1..1....1..1..1..1..1....0..0..0..0..0....0..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1104

Column 1 is A018910
Column 2 is A157727(n+3)
Column 3 is A187107(n+3)
Diagonal is A001595(n+2)
Superdiagonal 1 is A000071(n+5)
Superdiagonal 2 is A001610(n+3)
Superdiagonal 3 is A001595(n+4)
Superdiagonal 5 is A022308(n+5)
Superdiagonal 6 is A022319(n+5)
Superdiagonal 7 is A022407(n+5)
Superdiagonal 9 is A022323(n+7)

Empirical: T(n,k) = Fibonacci(n+3) +Fibonacci(k+3) -1

Empirical for rows, columns and nw-se diagonals: a(n) = 2*a(n-1) -a(n-3)

cp's OEIS Frontend

A121605 Primes of the form (Fibonacci[n+3] + 2) = A018910[n], Pisot sequence L(4,5).

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A121606 Indices n such that A018910[n] = (Fibonacci[n+3] + 2) are primes.

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A020717 Pisot sequences L(6,9), E(6,9).

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A020743 Pisot sequence L(7,10).

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A048584 Pisot sequence L(5,7).

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A166876 a(n) = a(n-1) + Fibonacci(n), a(1)=1983.

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A277084 Pisot sequence L(4,14).

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A258316 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with each 2X2 subblock having clockwise pattern 0000 or 0011.

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